An alternative to the Cauchy distribution.

A few generalizations of the Cauchy distribution appear in the literature. In this paper, a new generalization of the Cauchy distribution is proposed, namely, the exponentiated-exponential Cauchy distribution (EECD). Unlike the Cauchy distribution, EECD can have moments for some restricted parameters space. The distribution has wide range of skewness and kurtosis values and has a closed form cumulative distribution function. It can be left skewed, right skewed and symmetric. Two different estimation methods for the EECD parameters are studied. •A new generalization of the Cauchy distribution is proposed, namely, exponentiated-exponential Cauchy distribution (EECD).•EECD has flexible shape characteristics. Moreover, EECD moments are defined under some restrictions on the parameter space.

Specifications Table

Method details

The Cauchy distribution was first appeared in works of Pierre de Fermat and then studied by many researchers such as Isaac Newton, Gottfried Leibniz and others (see Ref. [1]). The Cauchy density was also used by Poisson [2] as counterexamples for some general results in probability. Based on Johnson et al. [1], the Cauchy distribution becomes associated with Cauchy [3] when Cauchy responded to an article by Bienayme’ [4] criticizing a method of interpolation proposed by Cauchy.

The fact that the Cauchy distribution has no moments, and therefore the law of large numbers does not apply, motivates researchers to generalize the Cauchy distribution. Few generalizations of the Cauchy distribution have appeared in the literature; Rider [5] proposed and study a generalization of the Cauchy distribution, Batschelet [6] proposed the wrapped-up Cauchy distribution, the skew-Cauchy distribution was proposed by Arnold and Beaver [7], another class of skew-Cauchy distribution was studied by Behboodian et al. [8], Huang and Chen [9] proposed a generalization of the skew-Cauchy distribution and recently Alshawarbeh et al. [10] used the beta family introduced by Eugene et al. [11] to generate the so called beta-Cauchy distribution.

In this paper, a new generalization of the Cauchy distribution is proposed. The proposed distribution is very flexible in terms of shapes, it can be left skewed, right skewed or symmetric. The moments are defined for some restricted values of the parameters. Also, the proposed distribution has a closed form cumulative distribution function (CDF) which adds more advantage to this distribution. The simplicity of the proposed distribution (closed from CDF) and the great flexibility in modeling real life data (see Application) will attract researchers to use this distribution as an alternative of the Cauchy distribution in modeling different scenarios.

Let be the probability density function (PDF) of a random variable , for . Let be a link function satisfies the following conditions:

The CDF of the T-X family of distributions defined by Alzaatreh et al. [12] is given bywhere satisfies the conditions in (1.1).

The corresponding PDF associated with (1.2) is

If , then satisfies the conditions (1.1) and (1.3) reduces towhere and are, respectively, the hazard and cumulative hazard functions associated with . Some general properties of the T-X in (1.4) have been recently studied, for more details see Alzaatreh et al. [12,13,31] and Lee et al. [14]. Also the discrete analogue of the T-X family is studied by Alzaatreh et al. [15].

If a random variable T follows the exponentiated exponential distribution (EED) with parameters and , , the definition in (1.4) leads to the exponentiated exponential-X family (EE-X) with the PDF

When and where n is a positive integer, the EE-X family in (1.5) reduces to the distribution of the first order statistics, , from a sample of size n from . When and , the EE-X family reduces to the distribution of the nth order statistics, , from a sample of size n from . When , the EE-X reduces to the exponentiated distribution with parameter . The parameters and controls the skewness and kurtosis of the family. Also, as , and as , .

The paper is outlined as follows. First we define using (1.5) a new generalization of the Cauchy distribution, namely, the exponentiated-exponential Cauchy (EEC) distribution. Then we study some properties of EEC distribution including quantile skeweness and kurtosis, Shannon entropy and moments. Also, different characterizations of the EE-X family based on truncated moments are discussed. Parameter Estimation deals with estimation methods of the EEC distribution. Applications of the EEC distribution to real data sets are provided.

The exponentiated-exponential Cauchy distribution

If X is a Cauchy random variable with parameter , then and , then (1.5) reduces to

A random variable X with the PDF g(x) in (2.1) is said to follow the exponentiated-exponential Cauchy distribution and will be denoted by EEC . When , the EEC distribution reduces to the exponentiated Cauchy distribution defined by Sarabia and Castillo [16]. When , the EEC distribution reduces to the Cauchy distribution with parameters . Therefore, the density in (2.1) is a generalization of the Cauchy density. From (2.1), we obtain the CDF of the EEC distribution as

The hazard function associated with the EEC distribution is

A physical interpretation of the EECD in (2.2) is possible for integer values of and . For example, let us start with a system which consists of independent components say . Suppose that each component consists of subcomponents. Assume that the system fails if all of the components fail (i.e. parallel system with respect to components). Assume further that each of the components fails if at least one of the subcomponents fails. Now, let X denote the lifetime of the system and be independent random variables follow Cauchy (). Then the CDF of X can be computed as follows

In the next section, some general properties of the EEC distribution will be addressed including transformations, limiting behavior, quantile function and the Shannon entropy.

The connection of EEC with some known distributions can be seen as follows; If a random variable Y follows Kumaraswamy’s distribution with parameters and , then the random variable follows EEC . Also, if a random variable Y follows beta distribution with shape parameters and , then the random variable follows EEC . Finally, if a random variable Y follows exponentiated-exponential distribution with parameters and , then the random variable follows EEC .

The mode of the EEC distribution is the solution of whereand . To find the mode of the EECD; first find such that and then obtain the mode at . If , then (2.4) implies that is the only mode. This result agrees with the mode of the standard Cauchy distribution.

If then EECD is unimodal.

To show EECD is unimodal, it suffices to show that in (2.4) has a unique solution on the interval (0, 1). First notice that and . Since is continuous on (0, 1), has at least one solution. Now let where and . Claim: on (0, 1). It is clear that and on (0, 1). Also, . The fact that implies that . Therefore, where . Furthermore, . Since and , we get . This implies that on (0,1). Now in order to show on (0,1), note that if we have . And since , and for all , we get . For the case , it is not difficult to show and hence on (0, 1). This ends the proof of the claim. Now,

. If and then . This implies that is strictly increasing and hence is strictly increasing function. Since has at least one solution, must have a unique solution on (0, 1)

It is not straightforward to show analytically that the equation has a unique solution for the case . However, from Fig. 1, Fig. 2 it appears that the EECD is a unimodal distribution.

Fig. 1

The EEC PDF for various values of α and .

Fig. 2

The EEC PDF for various values of α and .

In Fig. 1, Fig. 2, Fig. 3, various graphs of and are provided where the scale parameter . These plots indicate that the EEC PDF possesses great flexibility in terms of shapes; it can be symmetric, right skewed and left skewed. For fixed values of , the skewness (towards the left) of the distribution increases as increases. Also, for fixed values of , the skewness (towards the right) of the distribution increases as increases. Furthermore, Fig. 2 shows that the distribution is left skewed (right skewed) whenever () and . The plots in Fig. 3 indicate that the EEC hazard function shape is always upside-down bathtub.

Fig. 3

The EEC hazard function for various values of α and .

Let denote the quantile function for the EECD. Then, is given by

To explore the effect of the shape parameters when the quantile function is in closed form, Alzaatreh et al. [12] suggested using the quantile based Skewness and kurtosis for the T-X family of distributions. The measure of skewness S defined by Galton [17] and the measure of kurtosis K defined by Moors [18] are given by

When the distribution is symmetric, S = 0 and when the distribution is right (or left) skewed, S > 0 (or < 0). As K increases, the tail of the distribution becomes heavier. Since the CDF of the ECC distribution is in closed form, equations in (2.6) are used to obtain the Galtons' skewness and Moors' kurtosis where the quantile function is defined in (2.5). Fig. 4 displays the Galton's skewness and Moors' kurtosis for the EECD when . From Fig. 4, the EECD takes wide range of skewness and kurtosis values. This indicates that the EECD can be very effective in modeling real data sets with various skewness and kurtosis values.

Fig. 4

Galton’s Skewness and Moors’ Kurtosis for the EECD for various values of α and .

Galtons’ skewness is also used to determine the regions in which the EEC distribution is left skewed or right skewed. A numerical method is used to determine the points where the Galtons’ skewness equals to zero. Fig. 5 shows the regions in which the EEC distribution is left skewed or right skewed. The quadratic function in Fig. 5 connects the points where EEC distribution is symmetric.

Fig. 5

Skewness regions for the EECD when .

Some properties of EEC distribution

The entropy of a random variable X is a measure of variation of uncertainty [19].

Shannon entropy [20] for a random variable X with PDF g(x) is defined as . Since 1948, Shannon entropy has been used in many fields such as communication theory, engineering, physics and biology. Alzaatreh et al. [12] derived the Shannon entropy for the T-X family of distributions. Also, Ghosh and Alzaatreh [21] derived the Shannon entropy for the exponentiated exponential-X in (1.5) aswhere is the digamma function and T is the exponentiated-exponential random variable with parameters and . In the following theorem, we derive the Shannon entropy for the EEC distribution.where is the harmonic number of .

The Shannon entropy for the EEC distribution is given by

We first need to find , where and are, respectively, the PDF and the CDF of the Cauchy distribution. It is easy to show that and hence,

Now, using Gradshteyn and Ryzhik ([22], p. 55), can be written aswhere is the Bernoulli number.

By letting and using the series representation of in (3.4), (3.3) reduces towhere , the beta function. Now,

. The results in (3.1) followed by using the above result and substituting (3.5) in (3.2) and using the fact that . □

Moments

On using Remark 1, the rth moments for the EEC distribution can be written asBy using (3.4) and the fact that , one can obtain a series expansion for the as

Therefore, ([22], p. 17),where ,

Hence, from (3.6) the rth moments for the EEC distribution can be written as

The rth moments of the EEC do not always exist. The following theorem gives a necessary and sufficient condition for the existence of the rth moments of the EEC distribution.where is defined in (2.1).

The kth moments of the EEC distribution exist if and only if both and greater than k.

Consider the following integrals

Without loss of generality assume . Since the middle integral above exists, it suffices to investigate the existence of the first and third integrands. Let for . Then exists iff One can easily show that as and hence exists iff Similarly one can show exists iff  □

Fig. 6 provides the mean and variance of the EEC distribution when the scale parameter and for various combinations of and . From Fig. 6 it appears that for fixed , the mean is an increasing functions of while the variance is a decreasing function of . Also, for fixed , the mean is a decreasing function of .

Fig. 6

Mean and variance of the EECD for various values of α and .

The rth incomplete moments for the EEC distribution is defined as

On using the substitution , (3.10) can be written aswhere .

By using (3.8), (3.11) reduces towhere , is the incomplete beta function.

The first incomplete moment is used to find the deviations from the mean and median. The deviation from the mean and the deviation from the median are used to measure the dispersion and the spread in a population from the center. The mean deviation from the mean is denoted by , and the mean deviation from the median M is denoted by .where and can be found from Eq. (2.2).

The and for EEC distribution are

By definitions of and , it is easy to see that

and . The rest of the proof follows from (3.12). □

Some characterizations of the EE-X family based on truncated moments

Glänzel [25] provides characterizations based on the truncated moments for some important distributions including the standard Cauchy distribution. For more information, one is referred to Hamedani [26]. Below, we provide some results from Glänzel [25] which will be used to show Theorem 5.

Let be a continuous random variable and let and be two functions defined on for such that , where . Assume that , is of bounded variation and is continuously differentiable on H. Further assume that the equation has no solution on the interior of H. Finally, assume that . Then F is uniquely determined by the functions and . Furthermore, the density function of F is .

The following theorem provides a characterization for the EE-X family of distributions in (1.5).

Let be a continuous random variable. Then Y follows the EE-X family in (1.5) if and only if the functions in Theorem 4 can be chosen as: , and , , where F is the CDF of the random variable X defined in (1.5).

Using (1.5), one can show and . Therefore, . This implies that the equation has no solution on the interior of H. Also, it is not hard to show that . Since other assumptions of Theorem 4 are obvious, Y has the density function in (1.5). Furthermore, which implies that where the normalizing constant can be determine easily as . □

Let be a continuous random variable. Let , . Then Y follows the EE-X family in (1.5) if and only if there exist functions and satisfying the differential

Let be a continuous random variable. Then Y follows the EECD if and only if the functions in Theorem 4 can be chosen as: , and , , where for .

Let be a continuous random variable with CDF . Let be a differentiable function defined on H such that . Then for , if and only if .

See Domma and Hamedani [27]. □

Let be a continuous random variable. If and then Theorem 6 gives the CDF of the EE-X family in (1.5).

Parameter estimation

Maximum likelihood estimation method (MLE)

Let a random sample of size n be taken from the EEC distribution. The log-likelihood function for the EEC distribution in (2.1) is given bywhere .

The MLE , and can be obtained by maximizing the log likelihood function in (4.1) numerically. The initial value for is taken to be the MLE of by assuming the data, , follows the Cauchy distribution. The initial values for the parameters and are taken as follows: From Remark 1, the initial values of and are taken to be the MLEs of and by assuming the data follows . PROC NLMIXED in SAS is used to maximize the log-likelihood function in (4.1). In addition to the goodness of fit statistics, PROC NLMIXED gives the parameter estimates with their standard errors, which are the square roots of the diagonal entries in the estimated covariance matrix.

Alternative method of moment estimation (AMM)

Since the moments of the EECD do not always exist, we consider in this section an alternative method of moment estimation which was first proposed by Zografos and Balakrishnan [28].

If X follows EECD with parameters , and , then for any , , where is the CDF of the Cauchy distribution with parameter . □

Straightforward and hence omitted.

Using Theorem 7, and by equating the corresponding sample moments with the population moments we have the following three equations

The alternative method of moments estimates and are obtained by solving the equations in (4.2) iteratively.

Simulation study

To evaluate the performance of the MLE and the AMM methods, a simulation study for both methods is conducted for a total of five parameter combinations and the process is repeated 1000 times. Three different sample sizes n = 50, 70 and 100 are considered. The bias (estimate-actual), and the mean square errors (MSE) of the parameter estimates for the MLE and the AMM are presented in Table 1, Table 2 respectively. From the results in Table 2, it appears that the mean square errors for some parameters using the AMM method are unacceptably high. This can be seen more clearly for the parameter . The results in Table 1 show that the ML estimates, in most cases, have smaller mean square errors than the AMM estimates. Also, the bias using MLE method is acceptable. These results suggest using the MLE methods for data fitting. Also, a close look at the results from the small simulation study in Table 1, it is noticed that when (), the MLE of is overestimated (underestimated). Also, when (), the MLE of is overestimated (underestimated). Furthermore, Table 1 indicates that the MLE of is always overestimated.

Table 1

Bias and standard deviation of the parameter estimates using MLE method.

Sample size	Actual values			Bias			MSE
n	α	λ	θ	αˆ	λˆ	θˆ	αˆ	λˆ	θˆ
50	1	1	1	0.7997	0.2018	0.1785	0.7629	0.2460	0.2744
	1.5	0.5	1	−0.4319	0.4147	0.1799	0.6385	0.2669	0.2126
	1.5	1	2	−0.4231	0.7412	0.2556	0.6271	0.7634	0.5960
	0.8	0.5	0.7	0.3368	−0.0354	0.0945	0.1777	0.0138	0.0769
	0.6	1.2	2	0.4837	−0.4128	0.1791	0.2974	0.2052	0.4396
70	1	1	1	0.1334	0.1070	0.1010	0.1231	0.0874	0.1033
	1.5	0.5	1	−0.3842	0.3256	0.0908	0.2269	0.1462	0.1002
	1.5	1	2	−0.3760	0.6214	0.1448	0.2586	0.5113	0.3439
	0.8	0.5	0.7	0.2848	−0.0607	0.0557	0.1199	0.0111	0.0491
	0.6	1.2	2	0.4855	−0.4117	0.1766	0.2858	0.1981	0.3403
100	1	1	1	0.1111	0.0946	0.0691	0.0641	0.0523	0.0590
	1.5	0.5	1	−0.2347	0.3052	0.0514	0.2340	0.1205	0.0591
	1.5	1	2	−0.2369	0.5838	0.0825	0.2496	0.4299	0.2307
	0.8	0.5	0.7	0.2449	−0.0814	0.0261	0.0837	0.0112	0.0264
	0.6	1.2	2	0.4594	−0.4430	0.0866	0.2513	0.2184	0.2245

Table 2

Bias and standard deviation of the parameter estimates using AMM.

Sample size	Actual values			Bias			MSE
n	α	λ	θ	αˆ	λˆ	θˆ	αˆ	λˆ	θˆ
50	1	1	1	0.1709	0.0484	0.2244	0.6445	0.3925	0.9889
	1.5	0.5	1	−0.4835	0.6667	0.3531	0.7893	0.7783	1.3293
	1.5	1	2	0.1853	−0.0305	0.0519	1.6612	0.4617	2.8034
	0.8	0.5	0.7	−0.3502	0.5290	0.0932	0.1716	0.5116	0.2202
	0.6	1.2	2	0.0707	−0.3663	−0.2680	0.1972	0.4504	1.7952
70	1	1	1	0.1324	0.0026	0.1113	0.5628	0.2894	0.5619
	1.5	0.5	1	−0.6535	0.5351	0.1123	0.6377	0.5021	0.4480
	1.5	1	2	0.0709	−0.0544	0.0900	1.0081	0.3206	2.3761
	0.8	0.5	0.7	−0.3379	0.5733	0.1366	0.1534	0.5595	0.2098
	0.6	1.2	2	0.0457	−0.4095	−0.3439	0.2303	0.5065	1.3382
100	1	1	1	0.0625	−0.0463	0.0462	0.4782	0.2568	0.4878
	1.5	0.5	1	−0.6484	0.5646	0.1527	0.5515	0.5134	0.4186
	1.5	1	2	0.0022	−0.0853	−0.0700	0.7371	0.2513	1.2762
	0.8	0.5	0.7	−0.3850	0.4892	0.0609	0.1807	0.4212	0.1719
	0.6	1.2	2	−0.0120	−0.4529	−0.4426	0.1403	0.4586	1.3005

Application

To illustrate the applications of the EEC distribution, the EEC distribution is fitted to two data sets. The first data set in Table 3 (http://www.ibge.gov.br/seriesestatisticas/exibedados.php?idnivel=BR&idserie=PRECO101), is the INPC data which represents the national index of consumer prices of Brazil since 1979. The INPC index measures the cost of living of households with heads employees. The second data set in Table 4 from Weisberg [29], represents the sum of skin folds in102 male and 100 female athletes collected at the Australian Institute of Sports. The data sets are fitted to the EEC distribution and compared with the two-parameter Cauchy, the three parameter skew-Cauchy [8] and the beta-Cauchy [10] distributions. The following are the PDF of the Cauchy, skew-Cauchy and beta-Cauchy distributions respectively;

Table 3

The INPC data.

0.69	0.44	0.13	0.03	0.17	0.37	2.47	0.62	0.57	1.39	0.39
0.97	0.42	0.12	−0.11	0.50	0.39	2.70	0.31	0.84	0.30	0.55
0.43	0.49	0.27	0.70	0.73	0.82	3.39	1.07	0.48	−0.05	0.74
0.30	0.62	0.23	0.91	0.50	0.18	1.57	0.74	0.49	0.09	0.07
0.25	0.42	0.38	0.73	0.40	0.04	0.83	1.29	0.77	0.13	0.05
0.59	0.43	0.40	0.44	0.41	−0.06	0.86	0.94	0.55	0.05	0.47
0.32	0.16	0.54	0.57	0.57	0.99	1.15	0.44	0.29	0.61	1.28
0.31	−0.02	0.58	0.86	0.39	1.38	0.61	0.79	0.16	0.74	1.29
0.26	0.11	0.15	0.44	0.83	1.37	0.09	1.11	0.43	0.94	0.65
0.26	−0.07	0.00	0.17	0.54	1.46	0.68	0.60	1.21	0.96	0.42
−0.18	−0.28	0.49	0.15	0.18	0.68	0.34	1.20	0.29	1.51	2.46
0.11	0.15	0.54	0.29	0.35	0.45	0.38	1.33	0.71	1.40	2.18
−0.31	0.72	0.85	0.10	0.11	0.81	0.02	1.28	1.46	1.17	2.10
−0.49	0.45	0.57	−0.03	0.60	0.33	0.50	0.93	1.65	1.02	2.49
1.62	1.01	1.44

Table 4

The sum of skin folds data.

28.0	98	89.0	68.9	69.9	109.0	52.3	52.8	46.7	82.7	42.3
109.1	96.8	98.3	103.6	110.2	98.1	57.0	43.1	71.1	29.7	96.3
102.8	80.3	122.1	71.3	200.8	80.6	65.3	78.0	65.9	38.9	56.5
104.6	74.9	90.4	54.6	131.9	68.3	52.0	40.8	34.3	44.8	105.7
126.4	83.0	106.9	88.2	33.8	47.6	42.7	41.5	34.6	30.9	100.7
80.3	91.0	156.6	95.4	43.5	61.9	35.2	50.9	31.8	44.0	56.8
75.2	76.2	101.1	47.5	46.2	38.2	49.2	49.6	34.5	37.5	75.9
87.2	52.6	126.4	55.6	73.9	43.5	61.8	88.9	31.0	37.6	52.8
97.9	111.1	114.0	62.9	36.8	56.8	46.5	48.3	32.6	31.7	47.8
75.1	110.7	70.0	52.5	67	41.6	34.8	61.8	31.5	36.6	76.0
65.1	74.7	77.0	62.6	41.1	58.9	60.2	43.0	32.6	48	61.2
171.1	113.5	148.9	49.9	59.4	44.5	48.1	61.1	31.0	41.9	75.6
76.8	99.8	80.1	57.9	48.4	41.8	44.5	43.8	33.7	30.9	43.3
117.8	80.3	156.6	109.6	50.0	33.7	54.0	54.2	30.3	52.8	49.5
90.2	109.5	115.9	98.5	54.6	50.9	44.7	41.8	38.0	43.2	70.0
97.2	123.6	181.7	136.3	42.3	40.5	64.9	34.1	55.7	113.5	75.7
99.9	91.2	71.6	103.6	46.1	51.2	43.8	30.5	37.5	96.9	57.7
125.9	49.0	143.5	102.8	46.3	54.4	58.3	34.0	112.5	49.3	67.2
56.5	47.6	60.4	34.9

The maximum likelihood estimates, the log-likelihood value, the AIC (Akaike Information Criterion), the Kolmogorov-Smirnov (K-S) test statistic, and the p-value for the K-S statistic for the fitted distributions to the data sets are reported in Table 5, Table 6.

Table 5

Parameter estimates for the INPC data.

Distribution	Beta-Cauchy	EEC	Skew Cauchy	Cauchy
Parameter Estimates	cˆ = −0.0226 a(0.2279)	θˆ = 0.9949 (0.2891)	cˆ = 0.2424 (0.0818)	cˆ = 0.4792 (0.0323)
	θˆ = 0.7064 (0.1595)	αˆ = 27.3016 (18.5834)	θˆ = 0.3275 (0.0531)	θˆ = 0.2656 (0.0285)
	αˆ = 9.3393 (5.2784)	λˆ = 3.4706 (0.9507)	λˆ = 1.1888 (0.5256)
	λˆ = 4.0236 (1.6028)
ℓˆ	−116.5629	−116.7820	−132.7465	−139.3542
AIC	241.1257	239.5641	271.4929	282.7083
K-S	0.0376	0.0372	0.0837	0.1115
K-S p-value	0.9793	0.9815	0.2219	0.0403

Standard error.

Table 6

Parameter estimates for the sum of skin folds data.

Distribution	Beta-Cauchy	EEC	Skew Cauchy	Cauchy
Parameter Estimates	cˆ = 11.7939 (6.1381)	θˆ = 15.9717 (16.6657)	cˆ = 30.1404 (0.4983)	cˆ = 55.5789 (1.9777)
	θˆ = 19.4238 (6.1543)	αˆ = 666.6127 (128.0986)	θˆ = 27.9345 (2.6184)	θˆ = 16.9283 (1.6372)
	αˆ = 26.5961 (9.1507)	λˆ = 2.7858 (0.3756)	λˆ = 29.6768 (18.5649)
	λˆ = 4.0223 (1.0913)
ℓˆ	−955.0111	−955.7381	−972.6959	−1011.7310
AIC	1918.0220	1917.4760	1951.3920	2027.4630
K-S	0.0760	0.0700	0.1352	0.1794
K-S p-value	0.1937	0.2758	0.0012	0.0000

The results in Table 5, Table 6 show that the Cauchy distribution does not provide adequate fit to both data sets. The Skew-Cauchy distribution does not provide adequate fit to data sets in Table 4 and provides adequate fit to the data in Table 3. The EEC and beta-Cauchy distributions provide the best fit to the two data sets. The fact that EECD has less number of parameters compared with beta-Cauchy distribution makes EECD a better choice for fitting both data sets.

From Table 5, the estimated value for the parameter for the EEC distribution is approximately 1. Therefore, the two-parameter EEC distribution can be a natural choice for this data set. The likelihood ratio test for the hypothesis against confirms that the two-parameter EEC (standard EEC) distribution performs equally well when compared with the three-parameter EEC distribution. The results from fitting the standard EEC distribution to the INPC data as follows:

The likelihood ratio statistic in this case is based on , where and are the likelihood values for the standard EEC and the three-parameter EEC distributions respectively. The quantity asymptotically follows a chi-square distribution with 1 degree of freedom. In this case, we have and the p-value is 1.0000. Fig. 7 displays the empirical and the fitted cumulative distribution functions for the data sets in Table 3, Table 4. In this Figure, the standard EEC distribution is used for the data set in Table 3 and the three-parameter EEC distribution is used for the data set in Table 4. The figure supports the results from Table 5, Table 6.

Fig. 7

CDF for fitted distributions for data sets in Table 5, Table 6.

Summary and conclusion

In this article, a generalization of the Cauchy distribution, the EECD, is defined and studied. Several properties of the proposed distribution are studied in detail including mode, moments, skewness, kurtosis and Shannon entropy. Two real data sets are fitted to the EECD and compared with Cauchy, skew Cauchy and beta-Cauchy distributions. The results show the great flexibility of the proposed model. Based on Fig. 4, Fig. 5, the EEC indeed can fit different data sets with wide range of skewness values including left and right skewness. Furthermore, a simulation study is conducted for various parameter and sample size values to generate highly left and right skewed data sets from EECD. The results, based on K-S statistic, showed that the EECD produces good fit to various highly skewed (left and right) data sets. To conserve space, the results were not included in the article.

For future research, one can propose methods of discrimination between two or more members of the EE-X family based on the ratio of the Shannon entropies. For more information about this problem, one is referred to Zografos and Balakrishnan [28]. Furthermore, one can use the kullback-Leibler divergence [30] to discriminate between member of EE-X family and other family such as the beta family [11]. Also, it is noteworthy to mention that the method of discrimination between members of EE-X family using the idea proposed by Zografos and Balakrishnan [28] can be extended to cover the gamma-X family [13] or even the T-X family [12].

Subject Area:	Mathematics
More specific subject area:	Statistics, Distribution Theory
Method name:	The paper proposes an alternative to the Cauchy distribution using the T-X family framework proposed by Alzaatreh et al. (2013). The proposed distribution can be left skewed, right skewed or symmetric. The moments are defined under some restriction on the parameter space.
Name and reference of original method:	Alzaatreh, A., Lee, C. & Famoye, F. (2013). A new method for generatingfamilies of continuous distributions. Metron, 71, 63-79.
Resource availability:	Data source is mentioned in the paper